This section is intended to introduce various aspects of the art, which may be associated with embodiments of the disclosed techniques. This discussion is believed to assist in providing a framework to facilitate a better understanding of particular aspects of the disclosed techniques. Accordingly, it should be understood that this section is to be read in this light, and not necessarily as admissions of prior art.
Instruction limited algorithms may occur in various industries. For example, instruction limited algorithms may occur in areas such as 3D graphics analysis, encryption, data mining, compression, signal processing, image processing, chain rule evaluation, numerical methods such as finite element and finite volume analysis, seismic pattern recognition, and equation of state (EOS) calculations.
EOS calculations can be used to model phase behavior, which may have a significant effect on reservoir performance. The rate at which a petroleum mixture can flow through porous media is influenced by the number of phases, the viscosity of each phase, and the density of each phase. In general, phase behavior and phase properties are functions of temperature, pressure, and composition. In some cases, the compositional effects associated with phase behavior are weak and can be ignored. This may occur with petroleum fluids referred to as black oils. Modeling a reservoir containing black oils may be referred to as black oil simulation.
In other cases, the compositional effects may be accounted for. To account for compositional effects, the petroleum industry typically uses an equation of state (EOS). Modeling the reservoir with compositional effects may be referred to as a compositional simulation. A number of advanced recovery mechanisms can rely on the compositional effects of phase behavior.
EOS calculations may cause compositional simulations to be considerably slower than black oil simulations. EOS calculations typically involve determining the number of phases and the composition of each phase. Although an individual EOS calculation can be “cheap” to perform, it may be repeated billions of times over the course of a reservoir simulation. Thus, EOS calculations can consume more than 50% of total simulation time. Moreover, EOS calculations are computationally intensive and their cost may increase rapidly with the increase of the number of components in the mixture.
In order to improve computational speed, parallel reservoir simulators may be used. When using parallelization, a large problem is broken down into smaller subproblems, and then distributed between a number of processing cores. The subproblems may not be independent, and the cores can communicate to synchronize their work. The cores may communicate through shared memory or through high speed networks. In parallel computing environments, memory bandwidth and network communication are typical speed-limiting factors.
D. Voskov and H. Tchelepi, “Tie-Simplex Based Mathematical Framework for Thermodynamical Equilibrium Computation of Mixtures with an Arbitrary Number of Phases”, Fluid Phase Equilibria, Volume 283, 2009, pp. 1-11 states that a tie-line based parameterization method improves both the accuracy of the phase-behavior representation as well as the efficiency of equation of state (EOS) computations in compositional flow simulation. For immiscible compositional simulation, the technique is stated to use compositional space adaptive tabulation to avoid most of the redundant EOS calculations. However, matrix and vector operations are not optimized.
C. Rasmussen, et al., “Increasing the Computational Speed of Flash Calculations with Applications for Compositional, Transient Simulations”, SPE Reservoir Evaluation and Engineering, Volume 9, Number 1, 2009, pp. 32-38 states that in a conventional flash calculation, the majority of the simulation time is spent on stability analysis. The technique is stated to decide when it is justified to bypass the stability analysis, and does not optimize matrix and vector operations in instruction limited algorithms that perform EOS calculations.
E. Hendriks and A. Van Bergen, “Application of a Reduction Method to Phase-Equilibria Calculations”, Fluid Phase Equilibria, Volume 74, 1992, pp. 17-34 states that for specific thermodynamic models, the number of equations to solve a set of nonlinear equations relating to phase equilibrium of a mixture can be reduced. Problem size and the computational effort may be reduced through a variable transformation. Additionally, the smallest number of reduced variables that properly describe the phase behavior of the mixture may be determined. However, matrix and vector operations in instruction limited algorithms that perform EOS calculations are not optimized.